Holographic mode conversion for transmission lines

ABSTRACT

The present disclosure provides systems and methods associated with mode conversion for electromagnetic field modification. A mode converting structure (holographic metamaterial) is formed with a distribution of dielectric constants chosen to convert an electromagnetic radiation pattern from a first mode to a second mode to attain a target electromagnetic radiation pattern that is different from the input electromagnetic radiation pattern. A solution to a holographic equation provides a sufficiently accurate approximation of a distribution of dielectric constants that can be used to form a mode converting device for use with one or more transmission lines, such as waveguides. One or more optimization algorithms can be used to improve the efficiency of the mode conversion.

If an Application Data Sheet (ADS) has been filed on the filing date ofthis application, it is incorporated by reference herein. Anyapplications claimed on the ADS for priority under 35 U.S.C. §§ 119,120, 121, or 365(c), and any and all parent, grandparent,great-grandparent, etc., applications of such applications are alsoincorporated by reference, including any priority claims made in thoseapplications and any material incorporated by reference, to the extentsuch subject matter is not inconsistent herewith.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims the benefit of the earliest availableeffective filing date(s) from the following listed application(s) (the“Priority Applications”), if any, listed below (e.g., claims earliestavailable priority dates for other than provisional patent applicationsor claims benefits under 35 U.S.C. § 119(e) for provisional patentapplications, for any and all parent, grandparent, great-grandparent,etc., applications of the Priority Application(s)). In addition, thepresent application is related to the “Related Applications,” if any,listed below.

PRIORITY APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.14/708,043, filed May 8, 2017, for “Holographic Mode Conversion forTransmission Lines,” with inventors Tom Driscoll, John Desmond Hunt,Nathan Ingle Landy, David R. Smith, Yaroslav A. Urzhumov, whichapplication is incorporated herein by reference in its entirety.

RELATED APPLICATIONS

If the listings of applications provided above are inconsistent with thelistings provided via an ADS, it is the intent of the Applicant to claimpriority to each application that appears in the Priority Applicationssection of the ADS and to each application that appears in the PriorityApplications section of this application.

All subject matter of the Priority Applications and the RelatedApplications and of any and all parent, grandparent, great-grandparent,etc., applications of the Priority Applications and the RelatedApplications, including any priority claims, is incorporated herein byreference to the extent such subject matter is not inconsistentherewith.

TECHNICAL FIELD

This disclosure relates to dielectric mode converting structures fortransmission lines and waveguides. The dielectric mode convertingstructures are configured to convert electromagnetic energy from a firstmode to a second mode to modify one or more characteristics of theelectromagnetic energy.

SUMMARY

The present disclosure includes various systems, apparata, and methodsfor relating to mode converting structures configured to modifytransmitted electromagnetic energy. For example, a mode convertingstructure may have a volumetric distribution of dielectric constants tomodify a transmission in (or from) a waveguide or other transmissionline for a finite frequency range from a first mode to a second mode.The mode converting structure may be divided (actually and/orconceptually) into a plurality of sub-wavelength voxels. Each voxel mayhave a maximum dimension that is less than a wavelength within thefinite frequency range. Each voxel may be assigned one of a plurality ofdielectric constants to approximate a specific distribution ofdielectric constants of the mode converting structure.

As described in detail herein, any of a wide variety of methods andequations can be used to find a volumetric distribution of dielectricconstants, ε(x,y,z), given a desired or goal field distribution,E_(goal), and a measured, estimated, or otherwise known distribution ofelectromagnetic radiation (EMR) sources, Q(x,y,z).

Various methods of manufacturing are described herein, includingrotational molding, rotocasting, extrusion, and three-dimensionalprinting. Given a target transmission pattern for an EMR device anddomain boundaries for a mode converting structure, a mode convertingstructure can be generated that will convert the electromagnetic fieldgenerated by the EMR device from a first mode and field pattern to asecond mode and field pattern. The mode converting structure may bespecified as a volumetric distribution of dielectric constants that canbe approximated using a continuous manufacturing technique that involvesspatially inhomogeneous deposition of a homogeneous mixture of materialshaving various dielectric constants.

Relatedly, a dielectric structure may be divided into a plurality ofsub-wavelength voxels that each have a maximum dimension that is lessthan a wavelength (e.g., three-quarters, half, one-third, one-quarter,one-tenth of a wavelength) for a specific frequency range. Each voxelmay then be assigned one of a plurality of dielectric constants toapproximate an identified distribution of dielectric constants that willconvert electromagnetic energy from a first mode to a second mode for afirst waveguide. For example, the distribution of dielectric constantsmay convert electromagnetic energy within the waveguide at a first modeto a second mode.

As another example, the dielectric structure may be configured toconvert electromagnetic energy within a first waveguide from a firstmode to a second mode for transmission through a second (and optionallya third, fourth, fifth, etc.) waveguide. In some embodiments, thedielectric structure with a specific distribution of dielectricconstants may be configured to convert electromagnetic energy within afirst waveguide from a first mode to a second mode for transmission outof the waveguide into free space.

In each of the embodiments described herein, the various embodiments,modifications, adaptations, equations, algorithms, and/or othervariations may be adapted for use in free-space applications, intransmitting antennas, in receiving antennas, within a waveguide,between two different waveguides, from a transmission line to freespace, from a first transmission line to a second transmission line,between a waveguide transmission line to a non-waveguide transmissionline, from a non-waveguide transmission line to a waveguide transmissionline, for any of a wide variety of frequencies and bandwidths, and/or incombinations of any of the above.

Thus, embodiments in which EMR devices are described are equallyapplicable to embodiments relating to waveguides and other transmissionlines, even if not explicitly stated. Conversely, embodiments andvariations described in the context of waveguides and other transmissionlines are equally applicable to EMR devices for free-spaceelectromagnetic radiation transmission and reception. Additionally,embodiments described in the context of waveguides are equallyapplicable to various other transmission lines, and vice versa.

In at least some embodiments, the mode converting structure may bespecified as a volumetric distribution of dielectric constants to beapproximated using one or more discrete materials having specificdielectric constants. For example, a binary (two-levelpiecewise-constant) dielectric implementation can be used that is basedon the binary discretization of a calculated graded-index or continuousdistribution of dielectric constants. The dielectric constantdistribution may function as a holographic metamaterial for relevantfrequency range of an associated EMR device. The holographicmetamaterial concepts discussed herein should not be confused with“metamaterial holograms,” which relate to producing hologram images andare not capable of, or used for, converting the majority of the inputradiation into a mode with prescribed properties, as a means of creatingcustom electromagnetic field distributions in the near and/or far fieldzones.

The methods described herein provide a way to calculate a sufficientlyaccurate approximation of a volumetric distribution of dielectricconstants that will modify an input field from a first mode to a desiredoutput field in a second mode. Additionally, various manufacturingtechniques described herein, including a binary (or ternary, quaternary,etc.) three-dimensional printing approach, allow for a mode convertingstructure to be generated that sufficiently approximates the calculatedvolumetric distribution of dielectric constants.

Additional embodiments, variations, alternatives, and combinationsthereof are provided below. It is appreciated that any of the variousembodiments, alternatives, variations, features, and the like may becombined in any feasible and suitable way for a particular applicationand/or adaptation.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fee.

FIG. 1A illustrates an example of a volumetric holographic mediumshowing real values of a volumetric distribution of dielectric constantsfor increasing the directionality of a two-dimensionally isotropic linesource EMR device.

FIG. 1B illustrates an example of a holographic solution showingimaginary values of a volumetric distribution of dielectric constantsfor increasing the directionality of a two-dimensionally isotropic linesource EMR device.

FIG. 2A illustrates an example of a field distribution of a line sourceEMR device surrounded by a holographic metamaterial (mode convertingstructure) that has a distribution of dielectric constants approximatingthe distribution shown in FIGS. 1A and 1B.

FIG. 2B illustrates an example of a far-field radiation pattern of theline source in FIG. 2A surrounded by the mode converting structurehaving the distribution of dielectric constants shown in FIGS. 1A and1B.

FIG. 3 illustrates an example of a volumetric distribution of dielectricconstants for converting a radiation pattern of a two-dimensional pointdipole antenna into a directional beam.

FIG. 4A illustrates an example of a field distribution of atwo-dimensional point-dipole antenna surrounded by a mode convertingstructure that has a distribution of dielectric constants approximatingthe distribution shown in FIG. 3.

FIG. 4B illustrates an example of a far-field radiation pattern of thetwo-dimensional point dipole antenna in FIG. 4A surrounded by the modeconverting structure having the distribution of dielectric constantsshown in FIG. 3.

FIG. 5 illustrates an example of an optimized volumetric distribution ofdielectric constants generated by optimizing the solution shown in FIG.1A.

FIG. 6A illustrates an example of a field distribution of thetwo-dimensionally isotropic line source EMR device surrounded by a modeconverting structure that has a distribution of dielectric constantsapproximating the distribution shown in FIG. 5.

FIG. 6B illustrates an example of a far-field radiation pattern of thetwo-dimensionally isotropic line source EMR device in FIG. 6A surroundedby the mode converting structure having the distribution of dielectricconstants shown in FIG. 5.

FIG. 7 illustrates an example of an optimized volumetric distribution ofdielectric constants generated by optimizing the solution shown in FIG.3.

FIG. 8A illustrates an example of a field distribution of thetwo-dimensional point dipole antenna surrounded by a mode convertingstructure that has a distribution of dielectric constants approximatingthe distribution shown in FIG. 7.

FIG. 8B illustrates an example of a far-field radiation pattern of thetwo-dimensional point dipole antenna in FIG. 8A surrounded by the modeconverting structure having the distribution of dielectric constantsshown in FIG. 7.

FIG. 9A illustrates an example of a radiation intensity pattern of acircular horn antenna enhanced with an optimized binary mode convertingstructure.

FIG. 9B illustrates an example of a mode converting structure optimizedwith a binary volumetric distribution of dielectric constants configuredto be inserted into a horn antenna.

FIG. 9C illustrates the binary optimized mode converting structureinserted into the horn antenna.

FIG. 10 illustrates a directivity comparison of a typical horn antennaand an enhanced circular horn antenna with the binary optimized modeconverting structure insert in place.

FIG. 11A illustrates a representation of discretization of thevolumetric distribution of dielectric constants from FIG. 1A into aplurality of discrete dielectric constant values.

FIG. 11B illustrates a close up view of a portion of FIG. 11A.

FIG. 11C illustrates a representation of a possible embodiment of acylindrical mode converting structure with individual voxels assigneddiscrete dielectric constants.

FIG. 12 illustrates a representation of the effective distribution ofdielectric constants of the mode converting structure for voxels withsub-wavelength dimensions.

FIG. 13 illustrates one embodiment of a mode converting structure withina first waveguide configured to convert electromagnetic energy from afirst mode to a second mode.

FIG. 14A illustrates an embodiment of a mode converting structurepositioned between a first waveguide and a second waveguide.

FIG. 14B illustrates a simulated embodiment of electromagnetic energywith the first waveguide in a first mode converted by the modeconverting structure into a second mode within the second waveguide.

FIG. 15A illustrates an embodiment of a mode converting structurepositioned on the end of a first waveguide to convert electromagneticradiation from a first mode within the waveguide to a second mode forfree-space transmission.

FIG. 15B illustrates a simulated embodiment of electromagnetic energywithin the first waveguide in a first mode converted by the modeconverting structure into a second mode for free-space transmission.

FIGS. 16A-D illustrate various embodiments of waveguide junctions.

FIGS. 17A-D illustrate embodiments of waveguides joined using modeconverting structures to approximate and/or improve the functionality ofthe waveguides of FIGS. 16A-D.

FIG. 18A illustrates one embodiment of a mode converting structurepositioned inside of a waveguide to convert electromagnetic energy froma TE₀₁ mode, to a TM₁₁ mode.

FIG. 18B illustrates one embodiment of a mode converting structurepositioned inside of a waveguide to convert electromagnetic energy froma TE₁₀ mode, to a TM₁₁ mode.

FIG. 18C illustrates one embodiment of a mode converting structurepositioned inside of a waveguide to convert electromagnetic energy froma TE₁₁ mode, to a TM₁₁ mode.

DETAILED DESCRIPTION

According to various embodiments, systems, apparata, and methods aredescribed herein that relate to mode converting structures configured tomodify field patterns of electromagnetic radiation (EMR) devices. A modeconverting structure generated with a volumetric distribution ofdielectric constants can be used to convert an electromagnetic fieldfrom a first mode as generated by an original EMR device to a secondmode with more desirable properties. For example, the second mode mayhave a narrower beamwidth, a higher directional gain, lower far-fieldsidelobes, and/or a more uniform radiation profile in the radiativenear-field. In some embodiments, the mode converting structure maymodify the field pattern to compensate or negate the effects of are-radiating object in the near- or far-field of the EMR device.

The distribution of dielectric constants of the mode convertingstructure may be selected to modify a field pattern of the EMR devicefor a finite frequency range and convert EMR from a first mode to asecond mode. In various embodiments, the mode converting structure maybe idealized as a graded-permittivity structure having a continuousdistribution of dielectric constants, such that there are no abruptchanges in permittivity across the structure. Given a finite range ofwavelengths, a discretized piecewise-continuous approximation of thegraded-permittivity structure may be electromagnetically equivalent fora given bandwidth.

Thus, in various embodiments, the mode converting structure may bedivided into a plurality of sub-wavelength voxels. That is, the modeconverting structure may be conceptually thought of as comprising aplurality of voxels (three-dimensional pixels) whose largest dimensionis smaller than a wavelength within the relevant bandwidth. For example,each voxel may have a maximum dimension that is less than half of awavelength (e.g., the smallest wavelength) within a predeterminedfrequency range. The mode converting structure may be referred to as aholographic metamaterial device useful to modify the near-field and/orfar-field of an EMR device for a particular frequency range.

In some embodiments, the voxels may be cubes, parallelepipeds,tetrahedrons, prisms, various regular polyhedrons, or other polyhedrons.In some embodiments, a voxel may have one or two dimensions that aresub-wavelength while the other dimension(s) are larger than awavelength.

In various embodiments, a combination of voxel shapes and/or sizes maybe used. Moreover, voxels may be shaped and/or sized such that little orno space, gaps, or voids exist between voxels. Alternatively, voxels maybe arranged such that gaps or voids of various sizes and/or shapesexist. In some embodiments, the gaps or voids may be ignored and/ornegligible in calculating the volumetric dielectric constants.Alternatively, the gaps or voids may be assigned one or more dielectricconstants corresponding to a vacuum or to air or another fluid thatfills the gaps or voids.

Throughout this disclosure, discussions of discretizing the modeconverting structure may mean different things in various embodiments.For example, in some embodiments, the conceptual discretization of themode converting structure may be used for optimization algorithms, whilethe mode converting structure may not be physically discretized. In someembodiments, the discretization of the mode converting structure may bea physical discretization of the mode converting structure. Physicaldiscretization may be useful for manufacturing simplification (e.g., forthree-dimensional printing of a mode converting structure). Thus,allusions to discretizing, dividing into voxels, and the like should beunderstood in the context of either manufacturing or optimization, andsometimes both. In the absence of explicit context, discussions shouldbe construed as implying manufacturing and optimization individually,and as implying the possibility of a discretization for bothmanufacturing and optimization.

A manufactured mode converting structure may be positioned relative toan EMR device, may be removable, and/or may be configured as a retrofitsolution for an existing antenna system. The shape and dimensions of themode converting structure may be adapted based on the EMR device used.In various embodiments, an EMR device may include, by way of example butnot limitation, a radio frequency antenna, an optical radiationtransmitter, and an optical radiation receiver, and/or anelectro-optical EMR device configured to convert between electriccurrent and optical radiation or vice versa.

The mode converting structure may be used to modify field distributionsof the EMR device. For instance, a mode converting structure may be usedto modify the field distribution of a low-directivity antenna tocorrespond to that of a higher-directivity, narrow-beamwidth antenna.The mode converting structure allows for the beam pattern of the antennato be modified without any or at least any significant modification tometallic portions of the antenna, the antenna feed, and/or theenvironment of the antenna. Thus, the mode converting structuresdisclosed herein allow for a retrofit modification of existing antennasand/or other EMR devices.

The following specific examples use radio frequency (RF) antennas as anexample of EMR devices generally. However, it is appreciated that manyof the same concepts, embodiments, and general functionality of thesystems and methods described herein are equally applicable to otherfrequency ranges of EMR, including those utilizing low-frequency RF,microwave, millimeter-wave, Terahertz, far and mid-infrared, nearinfrared, visible light, ultraviolet, x-rays, gamma rays, and so forth.It is appreciated that the sizes, dielectric values, materials, andother variables may be adjusted based on the particular spectrum in use.

Many antennas, such as electrically small dipoles, monopoles, and loopantennas are fundamentally limited in their directionality. Horn-shapedantennas can have better directionality, so long as their dimensions arenot significantly sub-wavelength. At any rate, once fabricated andinstalled, fixed-shape, single-feed antennas generally have a fixedradiation pattern and a certain beamwidth, although it may vary based onfrequency.

Additionally, as described above, a dielectric structure may be dividedinto a plurality of sub-wavelength voxels that each have a maximumdimension that is less than a wavelength (e.g., three-quarters, half,one-third, one-quarter, one-tenth of a wavelength) for a specificfrequency range. As above, each voxel may then be assigned one of aplurality of dielectric constants to approximate an identifieddistribution of dielectric constants. Instead of being used forfree-space antennas for receiving and/or transmitting electromagneticradiation, the mode converting structure may be utilized to convertelectromagnetic energy from a first mode to a second mode within a firstwaveguide, between two waveguides, between a waveguide and free space,and/or between free space and a waveguide.

For example, the distribution of dielectric constants may convertelectromagnetic energy within the waveguide at a first mode to a secondmode. As another example, the dielectric structure may be configured toconvert electromagnetic energy within a first waveguide from a firstmode to a second mode for transmission through one or more additionalwaveguides.

Again, in each of the embodiments described herein, the embodiments,modifications, adaptations, equations, algorithms, and/or othervariations may be adapted for use in free-space applications, intransmitting antennas, in receiving antennas, within a waveguide,between two different waveguides, from a transmission line to freespace, from a first transmission line to a second transmission line,between a waveguide transmission line to a non-waveguide transmissionline, from a non-waveguide transmission line to a waveguide transmissionline, for any of a wide variety of frequencies and bandwidths, and/or incombinations and permutations of any of the above.

Thus, embodiments in which EMR devices, antennas, and free-spaceapplications are used as the example application for a mode convertingdevice are equally applicable to embodiments relating to waveguides andother transmission lines, even if not explicitly stated herein.

As described above, mode converting structures may be used to transformthe near-field and/or far-field of a fixed antenna without necessarilymodifying the antenna, installation, and/or surrounding environment.According to various embodiments, a holographic solution may be used todetermine a volumetric distribution of dielectric constants that canprovide a desired field transformation and mode conversion forfree-space applications and waveguide/transmission line applicationsalike.

For example, a volumetric distribution of dielectric constants can bedetermined using Equation 1 below, or a variation thereof:

ε_(hol)(x,y,z)−1=βE _(goal) ·E* _(n) /|E _(in)|²  Equation 1

In Equation 1, ε_(hol)(x,y,z) represents a volumetric distribution ofdielectric constants in an x, y, z coordinate system. In manyembodiments described herein, a Cartesian coordinate system is used as adefault example; however, any of a wide variety of coordinate systemsare suitable, including cylindrical, polar, barycentric, trilinear, andother coordinate systems. In fact, in some embodiments, alternativecoordinate systems may be preferable to simplify calculations and/orfacilitate manufacturing. For instance, a cylindrical coordinate systemmay be useful for a manufacturing technique in which the volumetricdistribution of dielectric constants corresponds to a uniform rotationof a two-dimensional planar cross section around an axis of revolution.

In Equation 1, represents a normalization constant and E_(in) representsan input field distribution of EMR from (1) an EMR device on the surfaceof the mode converting structure relative to the x, y, z coordinatesystem, (2) the input field distribution of EMR within a first waveguideor first waveguide portion relative to the x, y, z coordinate system, or(3) the input field distribution of the EMR from free space into awaveguide relative to the x, y, z coordinate system. Subsequentdescriptions of Equation 1 with application to EMR devices, such asantennas, are equally applicable to waveguide and free-spaceapplications. E_(goal) represents the “goal” or selected/desired outputfield distribution of EMR from the mode converting structure relative tothe x, y, z coordinate system.

The calculated distribution of dielectric constants may be approximatedby conceptually dividing the mode converting structure into a pluralityof voxels. Each voxel can then be assigned a permittivity value. In someembodiments, each voxel may be assigned a spatial average valuecorresponding to the average calculated permittivity value for thevolume of the voxel.

In embodiments in which the discretization is binary, ternary, or N-ary,each voxel may be assigned a dielectric constant from a selection of Ndiscrete dielectric constants, where N is an integer greater than 1 (2for binary, 3 for ternary, and so forth).

As a specific example, a region having a size that is distinguishable atthe frequency used by an EMR may contain multiple voxels. If the regionshould have, on average, a dielectric constant of 5.0, this may besatisfied by conceptually dividing the region into 100 voxels and usinggraphite, with a dielectric constant of 11, to fill 23 of those voxelsand polystyrene, with a dielectric constant of 3.2, to fill the other 77voxels. Thus, the average dielectric constant of the region willapproximate 5.0. Similar approximations can be made using any number ofmaterials having any number of dielectric constants. In someembodiments, frequency-dependent metamaterials having effectivedielectric constants less than 1.0 and exhibiting an active-gain can beused as well.

In various embodiments, the volumetric distribution of dielectricconstants may be substantially homogenous in one spatial dimension ofthe coordinate system, such that the volumetric distribution of the modeconverting structure is effectively two-dimensional even though it isphysically a three-dimensional object. For instance, the volumetricdistribution may correspond to a uniform extrusion of a planartwo-dimensional distribution perpendicular to the plane.

FIG. 1A illustrates an example of a holographic solution showing realvalues of a volumetric distribution of dielectric constants 100 usingEquation 1 above. The illustrated volumetric distribution of dielectricconstants is calculated for an idealized two-dimensionally isotropicline source EMR device. A mode converting structure (i.e., a holographicmetamaterial) with a corresponding distribution of dielectric constantscould be used as a cover for the line source EMR device to increase thedirectionality of the line source EMR device.

FIG. 1B illustrates the imaginary values of the volumetric distributionof dielectric constants 150 for the same line source EMR device usingEquation 1 above.

The “goal” or “target” field used in Equation 1 to generate FIGS. 1A and1B is a plane wave with infinite directivity. The example, althoughidealized, illustrates one method for generating a mode convertingstructure for converting the electromagnetic field generated by an EMRdevice to a second mode with improved radiation characteristics. Inpractice, the finite aperture of the holographic metamaterial domainlimits the actual directivity that can be attained. To account for theaperture effect, another option would be to use a Gaussian beam whosewaist is equal to or small than the diameter of the holographicmetamaterial domain.

FIG. 2A illustrates an example of a field distribution 200 of the linesource EMR device surrounded by a holographic metamaterial (modeconverting structure) that has a distribution of dielectric constantsapproximating the distributions (real and imaginary) shown in FIGS. 1Aand 1B.

FIG. 2B illustrates an example of a far-field radiation pattern 250 ofthe line source EMR device surrounded by the mode converting structurehaving the distribution of dielectric constants (real and imaginary)shown in FIGS. 1A and 1B.

Equation 1 above may result in a distribution of dielectric constantswith complex permittivity values in all four quadrants of the complexvariable plane, including the half-plane corresponding to active-gainmedium, and possibly the quadrant corresponding to a passive,negative-permittivity medium. In such embodiments, active-gainpermittivity values and negative permittivity values may be attainableusing metamaterials. For instance, the distribution of dielectricconstants can be discretized into sub-wavelength voxels each beingassigned a particular permittivity value. Some of the voxels may beassigned permittivity values that can be implemented with traditionallow-loss dielectrics, while other voxels may be assigned permittivityvalues (active-gain and negative) that can be implemented withmetamaterials.

In some situations, it may be desirable to utilize low-loss dielectricsin which ε≥1 and ε″<<1. Such materials may be referred to asnon-superluminal low-loss dielectrics (NSLLDs). Some material may onlybe considered NSLLD for specific frequency bands. Accordingly, thematerials used to generate a mode converting structure may depend highlyon the specific frequencies and bandwidths utilized by a particular EMRdevice.

According to various embodiments in which it is desirous to use NSLLDmaterials, a sufficiently accurate approximation to Equation 1 above isgiven by the equation below:

ε_(hol)(x,y,z)−1=β|E _(goal) +E _(in)|² /|E _(in)|²  Equation 2

In Equation 2 above, ε_(hol)(x,y,z) represents a volumetric distributionof dielectric constants in an x, y, z coordinate system. Again, anycoordinate system may be used that is suitable for the calculation ofthe distribution of dielectric constants and/or is useful for mapping amanufacturing process. represents a non-zero normalization constant andE_(in) represents an input field distribution of EMR from an EMR device,within a waveguide or other transmission line, and/or on a couplingbetween a waveguide and free space on the surface of the mode convertingstructure relative to the x, y, z coordinate system. E_(goal) representsthe “goal” or selected/desired output field distribution of EMR from themode converting structure relative to the x, y, z coordinate system.

Solving the equations above and/or other equations described herein, maybe performed using an optimization algorithm in which the dielectricconstants are treated as optimizable variables. The real and/orimaginary parts of the dielectric constants may be treated asindependently optimizable variables, or complex values may be selectedand used as the optimizable variables. Any of a wide variety ofoptimization algorithms may be used, including those (1) in which a costfunction is determined for each modification or group of modifications,(2) in which a gradient of a cost function based on partial derivativesis made with respect to each of the optimizable variable, and (3) inwhich a sensitivity vector is calculated using an adjoint sensitivityalgorithm.

In some embodiments, a constrained optimization algorithm may be used inwhich the dielectric constants are treated as optimization variablesconstrained to have real parts greater than or equal to approximately Nand imaginary parts equal to approximately M, where N and M are realnumbers. In other embodiments, a guess-and-check approach may be used inwhich an initial guess is used to solve the holographic solution usingany one of the equations described herein. Non-exhaustive examples ofspecific optimization algorithms are described in greater detail below.

FIG. 3 illustrates an example of a volumetric distribution of dielectricconstants 300 for converting a radiation pattern of a two-dimensionalpoint dipole antenna into a directional beam. The distribution ofdielectric constants illustrated in FIG. 3 is found using Equation 2above, with the target or goal output field set as a plan wave. For thepurposes of this calculation, the plane wave would be indistinguishablefrom a finite-width beam, given the finite diameter of the metamaterialdomain used.

FIG. 4A illustrates an example of a field distribution 400 of atwo-dimensional point-dipole antenna surrounded by a mode convertingstructure that has a distribution of dielectric constants approximatingthe distribution shown in FIG. 3.

FIG. 4B illustrates an example of a far-field radiation pattern 450 ofthe two-dimensional point dipole antenna in FIG. 4A surrounded by themode converting structure having the distribution of dielectricconstants shown in FIG. 3.

Equations 1 and 2 above provide adequate solutions to findingdistributions of dielectric constants for generating mode convertingstructures. However, further optimization may improve the efficiency ofthe mode conversion and compensate for the finite metamaterial domain.The metamaterial domain may be conceptually split into a plurality ofvoxels, where each voxel is approximately less than one-half wavelength(e.g., one-tenth of a wavelength). Each voxel may be conceptuallypopulated with a spatial average of the continuous dielectric constantfound using Equation 1 or 2 above. This discrete distribution ofdielectric constants may be used as an initial guess in an optimizationalgorithm. The optimization algorithm may treat the real and imaginaryvalues of the dielectric constant in each voxel as independent controlvariables. Alternatively, the complex (or real) value in each voxel maybe treated as an independent value.

Any of a wide variety of optimization algorithms may be used. Forexample, a small perturbation to one of the control variables may bemade, and then the forward wave propagation problem may be solved todetermine the effect of the perturbation. This may be referred to as acost function optimization in which the cost function is the differencebetween the target or goal field and the field produced by the currentstate of the optimization variables. The finite difference in the goal,divided by the small perturbation value of the control variable, may bereferred to as the finite-difference estimate of the cost functionpartial derivative. After computing all of the partial derivatives withrespect to all control variables, the combined vector may be referred toas the “gradient” of the cost function, also known as “a sensitivityvector.”

In other embodiments or as an alternative in the same embodiments, anadjoint method may be used that is based on the analytical derivativesof the equation describing the forward problem. The adjoint method maybe used to produce the entire sensitivity vector after solving just oneauxiliary problem known as the adjoint problem, whose computationalcomplexity is the same as the complexity of one forward problem of thesame size. In some applications, this may reduce the amount ofcomputation per optimization step by a factor of N, where N is thenumber of control variables.

Once a sensitivity vector is obtained, an iteration of a standardNewton, damped Newton, conjugate-gradient, or any other gradient-basediterative nonlinear solver may be used to determine the nextconfiguration.

In some embodiments the optimization algorithms may use heuristics aspart of the optimization process. This may be useful in embodimentswhere the control variables are non-differentiable. In those embodimentsit may be difficult to determine the gradient of the sensitivity vectoror the analytic derivative. For example, in some embodiments, theoptimization algorithm may use iterative heuristic optimizationtechniques such as, particle swarm optimization (PSO) or geneticoptimization to determine an optimal solution.

In one embodiment the iterative heuristic optimization may begin bygenerating a family of possible solutions. The family of possiblesolutions may contain thousands or millions of possible optimalsolutions. Each, of the possible solutions may be generated according toa predetermined representation of the optimal solution domain. In otherembodiments the predetermined solutions may be generated randomly. Thepredetermined representation may include properties of the optimalsolution (e.g., transmission mode, manufacturing constraints, dielectricproperties, boundary conditions, or optimization variables).

With each iteration of the heuristic optimization process the possiblesolutions may be evaluated using a fitness function. Solutions that are“more fit”, as determined by the fitness function, may be stochasticallyselected to continue on to the next iteration. Each, iteration may addnew possible solutions or remove less fit solutions. In addition, witheach successive iteration properties of each solution may be modified,altered, mutated, exchanged, updated, or changed in any way orcombination of ways useful in determining an optimal solution. Thealgorithm may iterate a finite number of times or it may iterate untilan acceptable solution is reached.

An optimization algorithm may be utilized until a predeterminedtermination tolerance(s) is met. A termination condition can be imposedon some norm of the sensitivity vector, in which case the optimizationalgorithm is guaranteed to converge. A termination condition can beimposed as an inequality on the scalar value of the cost function, inwhich case the algorithm may fail to meet the imposed condition. Invarious embodiments, it may be useful to apply a termination conditionto a sensitivity vector, and to take the final value of the optimizationcost function as an output of the algorithm instead of an input to thealgorithm.

For applications that require the final value of the cost function to bebelow a certain tolerance, the optimization loop that failed to producesuch an outcome can be repeated with a different initial guess. Each ofEquations 1 and 2 define a family of initial guesses, each of which canbe used to initiate a different optimization loop. Such loops areentirely independent and can be computed in parallel, using distributedcomputing.

FIG. 5 illustrates the optimization 500 of the real part of thedielectric constant distribution shown in FIG. 1A using the adjointsensitivity method and a conjugate-gradient nonlinear solver. In theillustrated optimized dielectric constant distribution, discretizedvalues from FIG. 1A are used as an initial guess. The optimizationalgorithm converged to the solution shown in FIG. 5.

FIG. 6A illustrates an example of a field distribution 600 of thetwo-dimensionally isotropic line source EMR device surrounded by a modeconverting structure that has a distribution of dielectric constantsapproximating the optimized distribution shown in FIG. 5.

FIG. 6B illustrates an example of a far-field radiation pattern 650 ofthe two-dimensionally isotropic line source EMR device surrounded by themode converting structure having the optimized distribution ofdielectric constants shown in FIG. 5. Comparison of FIG. 6A with FIG. 2Aand FIG. 6B with FIG. 2B show the improvement in mode conversionefficiency of the optimized solution.

FIG. 7 illustrates the optimization 700 of the dielectric constantdistribution shown in FIG. 3 using the adjoint sensitivity method and aconjugate-gradient nonlinear solver. In the illustrated optimizeddielectric constant distribution, discretized values from FIG. 3 areused as an initial guess. The optimization algorithm converged to thesolution shown in FIG. 7.

FIG. 8A illustrates an example of a field distribution 800 of thetwo-dimensional point dipole antenna surrounded by a mode convertingstructure that has a distribution of dielectric constants approximatingthe optimized distribution shown in FIG. 7.

FIG. 8B illustrates an example of a far-field radiation pattern 850 ofthe two-dimensional point dipole antenna in FIG. 8A surrounded by themode converting structure having the optimized distribution ofdielectric constants shown in FIG. 7. Comparison of FIG. 8A with FIG. 4Aand FIG. 8B with FIG. 4B show the improvement in mode conversionefficiency of the optimized solution.

The conceptual voxels described above are assigned a discretepermittivity value; however, the total number of unique values isunlimited as each one may be any real (or potentially complex) value. Insome embodiments, it may be useful to limit the total number of uniquevalues.

Thus, instead of assigning each voxel a value based on the spatialaverage, ε_(av), of the continuous distribution over that region, eachvoxel could be assigned, as a binary example, one of two values, ε₁ orε₂. For instance, each voxel may either be assigned a permittivity valueof “1” or “X”, where 1 represents a vacuum and X represents apermittivity value greater than 1. Such a binary discretization may bethought of as similar to gray-scale imaging where only white and blackdithering is used.

The Boolean decision to assign each voxel to either ε₁ or ε₂ may bebased on whether ε_(av) is above or below a threshold value. The resultmay be considered a piecewise-constant distribution of dielectricconstants. So long as the feature sizes of each voxel are sufficientlysmall (sub-wavelength at a minimum), the mode converting structure maybe electromagnetically equivalent to a continuous distribution for agiven bandwidth. In various embodiments, the piecewise-constantdistribution of dielectric constants may be binary, ternary, orquaternary in nature, or otherwise limited to a specific number ofunique permittivity values.

Thus, in some embodiments, Equation 1 and/or 2 may be used to determinea continuous distribution of dielectric constants. Optimizationalgorithms may then be employed using discretized average permittivityvalues. A mode converting structure may then be manufactured using thediscretized distribution of optimized average permittivity values.

In some embodiments, Equation 1 and/or 2 may be used to determine acontinuous distribution of dielectric constants. Optimization algorithmsmay then be employed using discretized average permittivity values. Thediscretized average permittivity values may then be discretized into Nvalues for an N-ary discretization (where N is 2 for binarydiscretization, 3 for ternary discretization, and so forth). A modeconverting structure may then be manufactured using the discretizeddistribution of optimized N-ary permittivity values. For example, anN-material three-dimensional printer may be used to deposit a materialwith one of the N permittivity values in each respective voxel.

In some embodiments, Equation 1 and/or 2 may be used to determine acontinuous distribution of dielectric constants. The continuousdistribution of dielectric constants may be discretized into N valuesfor an N-ary discretization (where N is 2 for binary discretization, 3for ternary discretization, and so forth). Optimization algorithms maythen be employed using the N-ary discretized average permittivityvalues. A mode converting structure may then be manufactured using thediscretized distribution of optimized N-ary permittivity values.

In embodiments in which the piecewise-constant distribution is used inthe optimization algorithms, it may be desirable to preserve the abilityto use real-valued control variables while still accounting for theN-ary nature of the structures being optimized. An algebraictransformation may be used to map the real-valued control variable tothe N-ary-valued dielectric constants. An example of such atransformation for a binary piecewise-constant distribution withpermittivity values ε₁ and ε₂ is as follows:

ε(x,y,z)=ε₁+(ε₂−ε₁)θ_(δ)(p(x,y,z))  Transformation 1

In Transformation 1, p(x,y,z) is a real-valued function of coordinateswith values bounded to the [−1; 1] interval (called the level-setfunction), and θ₅(p) is a smoothed Heaviside function, which, bydefinition, is equal to zero for p<−δ, unity for p>δ, and is continuouswith its first (and possibly second) derivatives for all p. The value ofthe smoothing parameter δ may be chosen as 0.1; however, this value canbe selected differently based on the specific application to achievemore accurate results.

The transformation allows optimization algorithms designed forcontinuous, real-valued control variables to be used for N-arydiscretized approximations by using near-N-ary values as realisticapproximations to N-ary values.

After optimization has been performed, the values may be converted backinto discretized N-ary values based on whether each optimized value isabove or below one or more threshold values, where the number ofthreshold values is equal to N−1.

FIG. 9A illustrates an example of a radiation intensity pattern 900 of acircular horn antenna 960 enhanced with an optimized binary modeconverting structure 970 (i.e., holographic metamaterial). Again, thegoal field can be set as a plane wave.

FIG. 9B illustrates an representation of a mode converting structure 970optimized with a binary volumetric distribution of dielectric constantsconfigured to be inserted into the horn antenna 960. The binaryvolumetric distribution of dielectric constants is illustrated asvarious grayscale patterns to show that average dielectric constantsover any given region may be a factor of the ratio of voxels assigned ε₁(shown as white) and ε₂ (shown as black). It can be appreciated that fora ternary or other N-ary embodiment, additional colors might be used torepresent the various possible discretization alternatives andapproximations.

FIG. 9C illustrates the binary optimized mode converting structure 970inserted into the horn antenna 960. As illustrated, the mode convertingstructure 970 may be specifically manufactured (i.e., a volumetricboundary may be imposed) so that it limits the total width to no widerthan the maximum width of the horn antenna 960. In the illustratedembodiment, the volumetric boundary allows the mode converting structure970 to protrude from the horn by a small amount.

FIG. 10 illustrates a directivity comparison 1000 of a typical hornantenna (shown as a solid line) and an enhanced circular horn antenna(shown as a dashed line) with the binary optimized mode convertingstructure insert in place. As illustrated, the binary optimized modeconverting structure provides a mode conversion with increaseddirectivity.

The equations above describe an x, y, z coordinate system. Many possiblevariations of Equations 1 and 2 are possible and may be utilized incombination with the discretization and optimization techniquesdescribed herein. A variation of Equation 1 that may be used is providedbelow:

ε_(hol)−1=|βE _(goal) ·E* _(in) /|E _(in)|²  Equation 3

In Equation 3 above, ε_(hol) represents a volumetric distribution ofdielectric constants in any of a wide variety of three-dimensionalcoordinate systems. Similar to Equation 1, β represents a normalizationconstant and E_(in) represents an input field distribution of EMR on amode converting waveguide junction (free space or other transmissionline) or from an EMR device on the surface of the mode convertingstructure relative to the three-dimensional coordinate system. E_(goal)represents a selected or desired output field distribution of EMR fromthe mode converting structure relative to the three-dimensionalcoordinate system.

In some embodiments, the holographic solutions to the equationsdescribed herein may be calculated with the electric field decomposedinto TE_(z) or TM_(z) mode in cylindrical coordinates and/or thedominant component of the TE_(z) or TM_(z) mode may be used in theholographic solution.

As previously described, the mode converting structure may be configuredand/or adapted for use with any of a wide variety of EMR devices,including but not limited to the following list of devices that are notnecessarily mutually exclusive: a short dipole antenna, a dipoleantenna, a horn antenna, a circular horn antenna, a metamaterial surfaceantenna technology (MSAT) device, a parabolic reflector, a monopoleantenna, a dipole antenna, a half-wave dipole antenna, a monopoleantenna, a folded dipole antenna, a loop antenna, a bowtie antenna, alog-periodic antenna, a slot antenna, a cavity-backed slot antenna, aninverted-F antenna, a slotted waveguide antenna, a waveguide, a Vivaldiantenna, a telescope, a helical antenna, a Yagi-Uda antenna system, aspiral antenna, a corner reflector, a parabolic reflector, a microstripantenna, and a planar inverted-F antenna (PIFA)

The mode converting structure may be adapted to electromagneticallyand/or mechanically couple to any of a wide variety of transmissionlines (where a waveguide is one type of transmission line). Examples ofsuch transmission lines include: hollow metal, circular pipe waveguides,rectangular waveguides, circular waveguides, elliptic waveguides,triangular waveguides, hexagonal waveguides, curved waveguides,dielectric waveguides, surface-wave waveguides, leaky waveguides,parallel lines, ladder transmission lines, twisted pair lines, star quadlines, coaxial cables, striplines, and microstrips.

In other embodiments, a mode converting structure may be adapted toelectromagnetically and/or mechanically couple one or more of: adielectric waveguide, a leaky dielectric waveguide, an optical fiber, amultimode optical fiber, a multicore optical fiber, a plasmonicwaveguide, a leaky plasmonic waveguide, a surface plasmon waveguide, andan optical polariton waveguide.

In some embodiments, a variation of Equation 2 may be used as providedbelow:

ε_(hol)−1=β|E _(goal) +E _(in)|² /|E _(in)|²  Equation 4

In Equation 4 above, ε_(hol) represents a volumetric distribution ofdielectric constants in any of a wide variety of three-dimensionalcoordinate systems. Similar to Equation 1, β represents a normalizationconstant and E_(in) represents an input field distribution of EMR on amode converting waveguide junction, at a waveguide termination, or froman EMR device on the surface of the mode converting structure relativeto the three-dimensional coordinate system. E_(goal) represents aselected or desired output field distribution of EMR from the modeconverting structure relative to the three-dimensional coordinatesystem.

In any of the embodiments described herein, values for ε_(hol) below aminimum threshold value may be set to a predetermined minimum value.Similarly, values for ε_(hol) above a maximum threshold value may be setto a predetermined maximum value. In other embodiments, a plurality ofdiscrete values for ε_(hol) may be available and each of the calculatedvalues of ε_(hol) may be assigned one of the available discrete valuesby rounding down to the nearest available value, rounding up to thenearest available value, and/or assigned to the closest matching value.

As previously described, many variations of Equations 1 and 2 may beused to find the holographic solution and calculate the volumetricdistribution of dielectric constants. Another example of such anequation is provided below:

ε_(hol)(x,y,z)=α+βE _(goal) ·E* _(in) /|E _(in)|²  Equation 5

In Equation 5 above, ε_(hol)(x,y,z) represents a volumetric distributionof dielectric constants in an x, y, z coordinate system. α and βrepresent selectable constants and E_(in) represents an input fielddistribution of EMR from an EMR device on the surface of the modeconverting structure relative to the x, y, z coordinate system. E_(goal)represents the “goal” or selected/desired output field distribution ofEMR from the mode converting structure relative to the x, y, zcoordinate system.

It is appreciated, without exhaustive recitation herein, that any of thevariations, embodiments, or methods for solving a holographic solutionor equation described herein may be used in conjunction with any of theother variations, embodiments, or methods of any other holographicsolution or equation.

In Equation 5, a value for a may be selected to optimize impedancematching between the input mode and a mode-converting medium. A value αmay be selected to optimize impedance matching between themode-converting medium and the output mode. A value α may be selected tomaintain a minimum value for ε_(hol) maintain ε_(hol) greater than 0, ormaintain ε_(hol) greater than 1.

As previously described, the mode converting structure may include oneor more metamaterials that have dielectric constants for a particularfrequency range. In some embodiments, physically small metamaterials maybe conglomerated to produce a sub-wavelength metamaterial conglomeratewith a specific dielectric constant.

The mode converting structure may be fabricated using any of a widevariety of materials. In many embodiments, the mode converting structuremay be purely dielectric in nature and/or may be composed substantiallyof NSLLDs. In other embodiments, the mode converting structure maycomprise substantially dielectric material or mostly dielectricmaterials. In still other embodiments, conductors may be utilized toachieve a particular output radiation pattern.

In various embodiments, a mode converting structure may be fabricatedwith a calculated distribution of dielectric constants using one or moreof amalgam compounding, material lamination, injection moldingprocesses, extrusion, foaming, compression molding, vacuum forming, blowmolding, rotational molding, casting, rotocasting, spin casting,machining, layer deposition, chemical etching, and dip molding. A modeconverting structure may be fabricated using one or more of a silica,polymers, glass-forming materials, a metamaterial, porcelain, glass,plastic, air, nitrogen, sulfur hexafluoride, parylene, mineral oil,ceramic, paper, mica, polyethylene, aluminum oxide, and/or othermaterial.

In various embodiments, an initial step may be to identify a targetfield pattern for an EMR device. Dimensional constraints may beidentified for a mode converting structure. For example, it may bedesirable that the mode converting structure have substantially the sameprofile or shape as the underlying EMR device. As a specific example, itmight be desirable that a mode converting structure fit into a cavity ofa horn antenna, as shown and described in conjunction with FIGS. 9A-9Cabove. In another embodiment, it might be desirable that the modeconverting structure be configured to replace or supplement an existingradome or protective cover associated with the EMR device. Any of a widevariety of volumetric constraints may be imposed. The mode convertingstructure can be manufactured to accommodate identified boundaries of athree-dimensional volume.

An input field distribution of EMR may be identified that will interactwith a surface of the mode converting structure. That is, an input fieldmay be identified at any number of points, planes, or other potentialsurfaces within the identified three-dimensional boundaries, withinwhich a generated mode converting structure is or may be positioned.

A mode converting structure can be manufactured that has the physicaldimensions that fit within the identified three-dimensional volume and avolumetric distribution of dielectric constants that will convert thefield to a second mode that approximates the target field pattern. Insome embodiments, the entity that makes the calculations, measurements,identifications, and determinations may be different from the entitythat actually manufactures the mode converting structure.

For example, a first entity may provide information to help inidentifying the target field pattern, the physical dimensions of adesired mode converting structure, and/or the input field distributionof EMR. A second entity may use this provided information to identifythe actual target field, input field, and dimensional constraints forthe purposes of the calculations. The second entity may then identify(i.e., calculate, estimate, and/or otherwise determine) a volumetricdistribution of constants.

The volumetric distribution of dielectric constants may be transmittedto the first party or a third party for manufacture of the modeconverting structure. Alternatively, the second party may alsomanufacture the mode converting structure. In still other embodiments, asingle party may perform all of the identification, determination, andmanufacturing steps. In short, any number of entities may perform anynumber of tasks and sub-tasks that aid in the manufacture of a modeconverting structure as described herein.

The distribution of dielectric constants may be a mathematicallycontinuous distribution, may be mathematically/conceptually divided intoa piecewise distribution (e.g., for optimization), and/or may bephysically divided into a piecewise distribution (e.g., formanufacturing). That is, the mode converting structure may be divided(conceptually and/or actually) into a plurality of sub-wavelengthvoxels. Each voxel may have one or more dimensions with a maximum thatis less than one half-wavelength in diameter for a specific frequencyrange. Each voxel may be assigned a dielectric constant based on thedetermined distribution of dielectric constants. Once manufactured, themode converting structure may convert the EMR from a first mode to asecond mode that approximates a target or goal field pattern.

FIG. 11A illustrates a representation 1100 of the discretization of thevolumetric distribution of dielectric constants from FIG. 1A into aplurality of discrete dielectric constant values. In the illustratedembodiment, the grayscale patterns in each of the boxes may eachrepresent one of N discrete permittivity values, in which case thevoxels are shown as relatively large for illustrative purposes.Alternatively, the grayscale patterns may represent a ratio ofunderlying binary permittivity values, in which case the individualboxes may represent averaged regions of tens, hundreds, or eventhousands of underlying voxels.

That is, FIG. 11A may be thought of as representing the distribution ofdielectric constants shown in FIG. 1A discretized into 29 uniquepermittivity values (see legend 1125) with a few hundred voxels in theentire image. Alternatively, legend 1125 may be thought of asrepresenting 29 possible ratios of permittivity values in a binarydiscretization with a few hundred regions shown in the image, in whicheach region comprises a plurality of underlying voxels whosepermittivity values have been averaged.

FIG. 11B illustrates a close up view 1150 of the representation ofindividual discrete voxels of the distribution shown in FIG. 11A.Assuming a binary discretization, each square in FIG. 11B may representan average of many underlying voxels.

FIG. 11C illustrates a representation of a possible embodiment of acylindrical mode converting structure 1130 with individual voxelsassigned discrete dielectric constants.

FIG. 12 illustrates a representation of the effective distribution ofdielectric constants of the mode converting structure of FIG. 11C forvoxels with sub-wavelength dimensions. As illustrated, if the featuresizes of each voxel are small enough, the discretized distribution ofdielectric constants closely approximates and may, for purposes of agiven bandwidth of an EMR device, be functionally equivalent to acontinuous distribution of dielectric constants. However, for theimplementation of optimization algorithms and/or to facilitate in themanufacturing process, it may be beneficial to discretize thedistribution of dielectric constants to include N discrete values, whereN is selected based on the manufacturing technique employed, the numberof available dielectric materials, and/or the homogenous orheterogeneous nature of such dielectrics.

One method of generating the mode converting structure comprises using athree-dimensional printer to deposit one or more materials having uniquedielectric constants. As described above, each voxel may be assigned adielectric constant based on the calculated distribution of dielectricconstants. The three-dimensional printer may be used to “fill” or“print” a voxel with a material corresponding to (perhaps equal to orapproximating) the assigned dielectric constant.

Three-dimensional printing using multiple materials may allow forvarious dielectric constants to be printed. In other embodiments, spacesor voids may be formed in which no material is printed. The spaces orvoids may be filled with a fluid or a vacuum, or ambient fluid(s) mayenter the voids (e.g., air).

In some embodiments, a multi-material three-dimensional printer may beused to print each voxel using a mixture or combination of the multiplematerials. The mixture or combination of multiple materials may beprinted as a homogeneous or heterogeneous mixture. In embodiments inwhich a homogeneous mixture is printed, the printer resolution may beapproximately equal to the voxel size. In embodiments in which aheterogeneous mixture is printed, the printer resolution may be muchsmaller than the voxel size and each voxel may be printed using acombination of materials whose average dielectric constant approximatesthe assigned dielectric constant for the particular voxel.

In some embodiments, the mode converting structure may be divided into aplurality of layers. Each of the layers may then be manufacturedindividually and then joined together to form the complete modeconverting structure. Each layer may, in some embodiments, be formed byremoving material from a plurality of voxels in a solid planar layer ofmaterial having a first dielectric constant.

The removed voxels may then be filled with material(s) having one ormore different dielectric constants. In some embodiments, the modeconverting structure may be rotationally symmetrical such that it can bemanufactured by creating a first planar portion and rotating it about anaxis.

As described above, a binary discretization may result in a plurality ofvoxels, each of which is assigned one of two possible permittivityvalues. The resolution and size of the voxels selected may be based onthe wavelength size of the frequency range being used.

In some embodiments, one of the two discrete dielectric constants may beapproximately 80. Another of the dielectric constants may beapproximately equal to a dielectric constant of distilled water at atemperature between 0 and 100 degrees Celsius. In some embodiments, oneof the two discrete dielectric constants and/or a third dielectricconstant may be approximately 1, such as air. As may be appreciated, theusage of a finite number of materials having a finite number of uniquedielectric constants and/or the usage of voxels having a non-zero sizemay result in a mode converting structure being fabricated that onlyapproximates the calculated continuous distribution of dielectricconstants.

Any of a wide variety of materials and methods of manufacturing may beemployed. For example, a mode converting structure may be manufactured,at least in part, using glass-forming materials, polymers,metamaterials, aperiodic photonic crystals, silica, compositemetamaterials, porous materials, foam materials, layered compositematerials, stratified composite materials, fiber-bundle materials,micro-rod materials, nano-rod materials, a non-superluminal low lossdielectric material, porcelain, glass, plastic, air, nitrogen, sulfurhexafluoride, parylene, mineral oil, ceramic, paper, mica, polyethylene,and aluminum oxide.

The mode converting structure may be fabricated by heating a materialabove a glass transition temperature and extruding a molten form of thematerial through a mask. The mask may be a rigid mask. Any otherfabrication method or combination of fabrication techniques may be used,including injection molding, chemical etching, chemical deposition,heating, ultrasonication, and/or other fabrication techniques known inthe art.

An NSLLD material may have a phase velocity for electromagnetic waves ata relevant frequency range that is less than c, where c is the speed oflight in a vacuum. Metamaterials may be used as effective media withdielectric constants less than 1 for a finite frequency range, and morethan one type or configuration of metamaterial may be used that hasunique dielectric constants. Various metamaterials may be used that havecomplex permittivity values. The complex permittivity values mayfunction as an effective-gain medium for a relevant frequency rangeand/or may correspond to a negative imaginary part of the effectivedielectric constant for the relevant frequency range.

The mode converting structure may be manufactured to have a width and/orlength similar to or corresponding to that of the EMR device. In variousembodiments, the mode converting structure may have a thickness that isless than one wavelength or a fraction of a wavelength of a frequencywithin a relevant frequency range for a particular EMR device. In otherembodiments, the mode converting structure may have a thicknessequivalent to several or even tens of wavelengths. The thickness of themode converting structure may be uniform or non-uniform and may besubstantially flat, rectangular, square, spherical, disc-shaped,parabolic in shape, and other have another shape or profile for aparticular application or to correspond to a particular EMR device.

In some embodiments, the mode converting structure may be configured tofunction as one of: an E-type T junction, an H-type T junction, a magicT hybrid junction, and a hybrid ring junction.

As previously described, the mode converting structure may bemanufactured to have a distribution of dielectric constants, or anapproximation thereof, that will cause a mode conversion of EMR outputby the EMR device from a first mode to a second mode through which atarget radiation pattern or “goal” may be attained. The target radiationpattern may, for example, be similar to that of an ideal half-wavedipole antenna with a directivity between approximately 2 and 5 dBi,that of an ideal horn antenna with a directivity of betweenapproximately 10 and 20 dBi, or that of an ideal dish antenna with adirectivity greater than 10 dBi.

As per the examples above, the mode converting structure may beconfigured to narrow the far-field beamwidth of the main lobe of the EMRdevice. In some embodiments, the distribution of dielectric constantsmay be calculated to create at least one deep minimum or null in afar-field directivity pattern.

In most embodiments, the mode converting structure may be configured toincrease the directional gain of the EMR device. In some embodiments,the mode converting structure may be configured to perform one or morefunctions, including: decrease maximum sidelobe lever; decreasefar-field sidelobes; decrease directivity in one or more directions;decrease the power of at least one sidelobe; decrease the power radiatedinto a specific solid angle; change the direction of a strongestsidelobe; change the direction of a sidelobe closest to a boresight;decrease radiation in an approximately opposite direction of a main lobedirection; decrease radiation in a backward half space defined as thedirection between approximately 180 and 270 degrees relative to theboresight; decrease radiation with a selected polarization; change apolarization of at least some of the radiated EMR from a firstpolarization to a second polarization; increase the uniformity of theradiation profile of the EMR device in the near-field; create a null ofelectric field in the near-field of the EMR device; create a null ofmagnetic field in the near-field of the EMR device; create aconcentration of electromagnetic energy density of electric field in thenear-field of the EMR device; create a concentration of energy densityof magnetic field in the near-field of the EMR device; reduce peakvalues of electric field in the near-field of the EMR device; and/orreduce peak values of magnetic field in the near-field of the EMRdevice.

In some embodiments, the mode converting structure may be configuredwith a distribution of dielectric constants to modify the far-fieldradiation pattern to compensate for a re-radiating object in the near-or far-field of the EMR device. For example, support structures,interfering objects, structures, vehicles, other antennas, and/ormetallic objects may be in the near-field of an antenna system andimpact the far-field radiation pattern. Accordingly, a mode convertingstructure may be adapted to specifically modify the far-field pattern tomake it as if the object(s) in the near-field were substantially absent.

In various embodiments, a mode converting structure may include adistribution of discretized dielectric constants configured for use witha planar antenna. The planar antenna may be housed within a radomeand/or a protective casing. The mode converting structure may be part ofthe radome and/or the protective casing, or applied to the radome and/orthe protective casing after installation. The mode converting structuremay modify the far-field and/or near-field radiation pattern of theplanar antenna and/or compensate for any re-radiating objects innear-field of the planar antenna.

In other embodiments, a mode converting structure may be formed as partof a radome for a horn antenna. The radome may include additionalcomponents or features to secure the mode converting structure to asupport of the horn antenna. In still other embodiments, a modeconverting structure may be formed as part of a protective cover for adipole antenna.

In each of FIGS. 13-17D, the illustrated grayscale shading of the modeconverting structures 1375, 1475, 1575, 1700, 1710, 1720, and 1730 isfor illustrative purposes only and does not correspond to a usefuldistribution of dielectric constants and is not intended to represent anactual or even plausible distribution of dielectric constants. Actualdimensions and distributions of dielectric constants can be calculatedand/or optimized using the various algorithms, methods, and approachesdescribed herein. Moreover, while the illustrated embodiments show anddescribe various applications of the above-described systems and methodsusing waveguides as examples, it is appreciated that the various systemsand methods described herein are applicable to a wide variety oftransmission lines and not just waveguides.

FIG. 13 illustrates one embodiment 1300 of a mode converting structure1375 within a first waveguide 1310 configured to convert EMR from afirst mode 1350 to an EMR with a second mode 1350′. The mode convertingstructure 1375 may be said to couple a first waveguide with EMR in thefirst mode 1350 and a second waveguide with EMR in the second mode1350′. Alternatively, the mode converting structure 1375 may bedescribed as an insert or component within a single waveguide. Asillustrated, the first portion of the waveguide (or first waveguide)with the EMR in the first mode 1350 may have different dimensions thanthe second portion of the waveguide (or second waveguide) with the EMRin the second mode 1350′.

FIG. 14A illustrates an embodiment 1400 of a mode converting structure1475 positioned between a first waveguide 1410 and a second waveguide1420. As illustrated, the mode converting structure 1475 may havedimensions in one or more directions that exceed that of one or bothwaveguides 1410 and 1420. In various embodiments, the mode convertingstructure 1475 may mechanically, magnetically, and/or otherwise couplethe first waveguide to the second waveguide. In some embodiments, themode converting structure 1475 may be inserted into a waveguide coupleconfigured to couple two waveguides together. In still otherembodiments, the mode converting structure 1475 may be inserted within acoupling device configured to join multiple sections of a waveguidetogether.

FIG. 14B illustrates a simulated embodiment 1401 of EMR in a first mode1450 with the peak electric field concentrated near the perimeter(darker shading). A mode converting structure 1475 converts the EMR fromthe first mode 1450 with a peak electric field concentrated near theperimeter of the first waveguide 1410 to EMR in a second mode 1450′ withthe peak electric field concentrated near the center of the secondwaveguide 1420 (again, darker shading).

FIG. 15A illustrates an embodiment 1500 of a mode converting structure1575 positioned on the end of a first waveguide 1510 to convert EMR froma first mode within the waveguide 1510 to a second mode for free-spacetransmission.

FIG. 15B illustrates a simulated embodiment 1501 of EMR within a firstwaveguide 1510 in a first mode 1550 converted by a mode convertingstructure 1575 into a second mode 1550′ for free-space transmission. Themode converting structure 1575 may be adapted to focus or,alternatively, disperse the EMR for free-space transmission. In someembodiments, the mode converting structure 1575 may be adapted totransfer a signal to another waveguide over a short free-space gap. Inother embodiments, the mode converting structure 1575 may be used totransmit EMR in the second mode 1550′ via free space to a receivingantenna.

FIG. 16A illustrates an E-type T junction 1600 with the top of the “T”1602 extending from the main waveguide (1601 through to 1603) in thesame direction as the electric field. In various embodiments, the outputwaveguide 1603 is 180° out of phase with respect to the input waveguide1601.

FIG. 16B illustrates an H-type T junction 1610 with the long axis of thewaveguide (extending from 1601 to 1603) is parallel to the plane of themagnetic lines of force within the waveguide. In some embodiments, theH-type T junction may be used to connect waveguides while preservingphase regardless of which ports 1601, 1603, and 1604 are used.

FIG. 16C illustrates a Magic T hybrid waveguide junction 1620 combiningan H-type and E-type T junction. When a signal is applied to port 1604,no signal appears at port 1602 and the two signals appearing at ports1601 and 1603 are 180° out of phase with respect to each other. When asignal is input via port 1601, a signal appears at ports 1602 and 1604,but not at port 1603. When a signal is input via port 1603, a signalappears at ports 1602 and 1604, but not at port 1601.

FIG. 16D illustrates a hybrid ring waveguide junction 1630. In someembodiments, port 1605 may be connected to an antenna, port 1606 may beconnected to a receiver, port 1607 may be connected to a transmitter,and port 1608 may be connected to a receiver. The hybrid ring waveguidejunction 1630 may function as a duplexer.

During a transmit period/cycle, the hybrid ring waveguide junction 1630may couple the transmitter port 1607 to the antenna port 1605 withoutenergy being conveyed to the receiver ports 1606 and 1608. During thereceive period/cycle, the hybrid ring waveguide junction 1630 may coupleenergy from the antenna port 1605 to the receiver 1606 and/or 1608without energy being conveyed to the transmitter port 1607.

FIG. 17A illustrates a mode converting structure 1700 configured with upto three ports 1601, 1602, and 1603 for connecting up to threewaveguides. The mode converting structure 1700 may be configured with adistribution of dielectric constants configured to function similar toan E-type T junction, even if one or more ports is not actually present.For example, mode converting structure 1700 may connect waveguides viaports 1601 and 1603 180° out of phase without any waveguide connected toport 1602 or even without port 1602.

FIG. 17B illustrates a mode converting structure 1710 configured with upto three ports 1601, 1602, and 1603 for connecting up to threewaveguides. The mode converting structure 1710 may be configured with adistribution of dielectric constants configured to function similar toan H-type T junction, even if one or more ports is not actually present.For example, mode converting structure 1710 may connect waveguides viaany two or three of ports 1601, 1603, and 1604.

FIG. 17C illustrates a mode converting structure 1720 configured with upto four ports 1601, 1602, 1603, and 1604 for connecting up to fourwaveguides. The mode converting structure 1720 may be configured with adistribution of dielectric constants configured to function similar to aMagic T hybrid waveguide junction, even if one or more ports is notactually present and/or not connected to a waveguide.

For example, regardless of which ports are present and/or how many portsare actually connected to mode converting structure 1720: (1) when asignal is applied to port 1604, the signals may appear at ports 1601 and1603 that are 180° out of phase with respect to each other; (2) when asignal is input via port 1601, a signal may appear at ports 1602 and1604; and (3) when a signal is input via port 1603, a signal appears atports 1602 and 1604.

FIG. 17D illustrates a mode converting structure 1730 configured with upto four ports 1605, 1606, 1607, and 1608 for connecting up to fourwaveguides. The mode converting structure 1730 may be configured with adistribution of dielectric constants configured to function similar to ahybrid ring waveguide junction as described in FIG. 16D, even if one ormore ports is not actually present and/or not connected to a waveguide.

FIG. 18A illustrates a mode converting structure 1875 positioned insideof a waveguide 1800. The mode converting structure 1875 may beconfigured with a distribution of dielectric constants configured toconvert EMR from one mode to another.

For example, the mode converting structure may convert EMR from a firstmode 1850 to a second mode 1850′.

As a specific example, the first mode 1850 may comprise EMR in a TE₀₁mode, where the electric field is transverse to the direction ofpropagation. In the TE₀₁ mode 1850 the wide dimension of the waveguide1800 is one half of the wavelength of the electric field and the narrowdimension is less than one half of the wavelength of the magnetic field.The second mode 1850′ may be EMR in a TM₁₁ mode, where the magneticfield is transverse to the direction of the direction of propagation. Inthe second mode 1850′, the wide direction of the waveguide is equal tohalf the wavelength of the magnetic field and the narrow direction ofthe waveguide is equal to half the wavelength of the electric field.

In the illustrated embodiments, the mode converting structure 1875changes the mode of the EMR from first mode to a second mode. In someembodiments, the mode converting structure 875 may the mode of the EMRand modify one or more properties of the EMR such as, but not limitedto, the wavelength, polarization, frequency, amplitude, phase, and/orfocus).

In an alternative embodiment, the mode converting structure 1875 may bejuxtapositioned between two different waveguides to the left and rightof the mode converting structure 1875, as opposed to being an insertwithin a single, continuous waveguide.

Additionally, in some embodiments the waveguide portion (or distinctwaveguide) to the left of the mode converting structure 1875 may be adifferent dimension than the waveguide portion (or distinct waveguide)to the right of the mode converting structure 1875. Similarly, thewaveguide or waveguide portion housing the mode converting structure1875 may have a different dimension than either of the other waveguidesor waveguide portions.

FIG. 18B illustrates a waveguide 1801 containing a mode convertingstructure 1876. The mode converting structure 1876 may be configuredwith a distribution of dielectric constants configured to convert EMRfrom a first mode 1851 to a second mode 1851′. The first mode may beTE₁₀ and the second mode may be TM₁₁. The mode converting structure 1876may be configured to convert EMR in a TE_(x,y) mode into EMR in aTM_(m,n) mode, where m, n, x, and y are non-negative integers.

FIG. 18C illustrates a waveguide 1802 containing a mode convertingstructure 1877. The mode converting structure 1877 may be configuredwith a distribution of dielectric constants configured to convert EMRfrom a first mode 1853 to a second mode 1853′. The first mode may be, asa specific example, TE₁₁ and the second mode may be TM₁₁. Similar toFIG. 18A, the mode of the EMR may be changed from transverse electric totransverse magnetic, or vice versa.

Many existing computing devices and infrastructures may be used incombination with the presently described systems and methods. Some ofthe infrastructure that can be used with embodiments disclosed herein isalready available, such as general-purpose computers, computerprogramming tools and techniques, digital storage media, andcommunication links. A computing device or controller may include aprocessor, such as a microprocessor, a microcontroller, logic circuitry,or the like. A processor may include a special purpose processingdevice, such as application-specific integrated circuits (ASIC),programmable array logic (PAL), programmable logic array (PLA),programmable logic device (PLD), field programmable gate array (FPGA),or other customizable and/or programmable device. The computing devicemay also include a machine-readable storage device, such as non-volatilememory, static RAM, dynamic RAM, ROM, CD-ROM, disk, tape, magnetic,optical, flash memory, or other machine-readable storage medium. Variousaspects of certain embodiments may be implemented using hardware,software, firmware, or a combination thereof.

The components of the disclosed embodiments, as generally described andillustrated in the figures herein, could be arranged and designed in awide variety of different configurations. Furthermore, the features,structures, and operations associated with one embodiment may beapplicable to or combined with the features, structures, or operationsdescribed in conjunction with another embodiment. In many instances,well-known structures, materials, or operations are not shown ordescribed in detail to avoid obscuring aspects of this disclosure.

The embodiments of the systems and methods provided within thisdisclosure are not intended to limit the scope of the disclosure, butare merely representative of possible embodiments. In addition, thesteps of a method do not necessarily need to be executed in any specificorder, or even sequentially, nor do the steps need to be executed onlyonce. As described above, descriptions and variations described in termsof transmitters are equally applicable to receivers, and vice versa.

This disclosure has been made with reference to various exemplaryembodiments, including the best mode. However, those skilled in the artwill recognize that changes and modifications may be made to theexemplary embodiments without departing from the scope of the presentdisclosure. While the principles of this disclosure have been shown invarious embodiments, many modifications of structure, arrangements,proportions, elements, materials, and components may be adapted for aspecific environment and/or operating requirements without departingfrom the principles and scope of this disclosure. These and otherchanges or modifications are intended to be included within the scope ofthe present disclosure.

This disclosure is to be regarded in an illustrative rather than arestrictive sense, and all such modifications are intended to beincluded within the scope thereof. Likewise, benefits, other advantages,and solutions to problems have been described above with regard tovarious embodiments. However, benefits, advantages, solutions toproblems, and any element(s) that may cause any benefit, advantage, orsolution to occur or become more pronounced are not to be construed as acritical, required, or essential feature or element. The scope of thepresent invention should, therefore, be determined by the followingclaims.

1. (canceled)
 2. An electromagnetic mode converting structurecomprising: a dielectric structure configured to modify a field patternof a device from a first mode to a second mode, the dielectric structuredivided into a plurality of sub-wavelength voxels, wherein each voxelhas a maximum dimension that is less than a wavelength for a specificfrequency range, and each voxel is assigned one of a plurality ofdielectric constants to approximate the distribution of the dielectricconstants.
 3. The electromagnetic mode converting structure of claim 2,wherein the dielectric structure includes real and imaginary parts, eachof the real and imaginary parts being individually optimizablevariables.
 4. The electromagnetic mode converting structure of claim 3,wherein real parts are greater than or equal to approximately one andthe imaginary parts are equal to approximately zero.
 5. Theelectromagnetic mode converting structure of claim 2, wherein theapproximation of the distribution of the dielectric constant furthercomprises modification of at least one dielectric constant and determinea cost function for the modification.
 6. The electromagnetic modeconverting structure of claim 2, wherein the approximation thedistribution of the dielectric constants further comprises a holographicsolution.
 7. The electromagnetic mode converting structure of claim 2,wherein the distribution of dielectric constants is a three-dimensionalcoordinate systems.
 8. The electromagnetic mode converting structure ofclaim 7, wherein the three-dimensional coordinate systems is one of aCartesian coordinate system or a cylindrical coordinate system.
 9. Theelectromagnetic mode converting structure of claim 2, wherein the firstmode is a TE₀₁ mode and the second mode is a TM₁₁ mode.
 10. Theelectromagnetic mode converting structure of claim 2, wherein the firstmode is a TE₁₀ mode and the second mode is a TM₁₁ mode.
 11. Theelectromagnetic mode converting structure of claim 2, wherein the firstmode is a TE₁₁ mode and the second mode is a TM₁₁ mode.
 12. Theelectromagnetic mode converting structure of claim 2, wherein theelectromagnetic mode converting structure is dimensionally approximateto the device.
 13. The electromagnetic mode converting structure ofclaim 2, further comprises a metamaterial with an effective dielectricconstant less than 1 for at least a portion of a finite frequency range.14. A method of generating an electromagnetic mode converting structure,the method comprising: generating, using one or more materials havingdielectric constants, an electromagnetic mode converting structure, theelectromagnetic mode converting structure comprising: a dielectricstructure configured to modify a field pattern of a device from a firstmode to a second mode, the dielectric structure divided into a pluralityof sub-wavelength voxels, wherein each voxel has a maximum dimensionthat is less than a wavelength for a specific frequency range, and eachvoxel is assigned one of a plurality of dielectric constants toapproximate the distribution of the dielectric constants.
 15. The methodof claim 14, wherein the generation of the electromagnetic modeconverting structure is performed by a three-dimensional printer. 16.The method of claim 14, wherein the generation of the electromagneticmode converting structure is performed by injection molding.
 17. Themethod of claim 14, wherein the generation of the electromagnetic modeconverting structure is performed by one of chemical etching, chemicaldeposition, heating or ultrasonication.
 18. The method of claim 14,wherein the dielectric structure includes real and imaginary parts, eachof the real and imaginary parts being individually optimizablevariables.
 19. The method of claim 18, wherein real parts are greaterthan or equal to approximately one and the imaginary parts are equal toapproximately zero.
 20. The method of claim 14, wherein theapproximation of the distribution of the dielectric constant furthercomprises modification of at least one dielectric constant and determinea cost function for the modification.
 21. The method of claim 14,wherein the approximation the distribution of the dielectric constantsfurther comprises a holographic solution.
 22. The method of claim 14,wherein the distribution of dielectric constants is a three-dimensionalcoordinate systems.
 23. The method of claim 22, wherein thethree-dimensional coordinate systems is one of a Cartesian coordinatesystem or a cylindrical coordinate system.
 24. The method of claim 14,wherein the first mode is a TE₀₁ mode and the second mode is a TM₁₁mode.
 25. The method of claim 14, wherein the first mode is a TE₁₀ modeand the second mode is a TM₁₁ mode.
 26. The method of claim 14, whereinthe first mode is a TE₁₁ mode and the second mode is a TM₁₁ mode. 27.The method of claim 14, wherein the electromagnetic mode convertingstructure is dimensionally approximate to the device.
 28. The method ofclaim 14, further comprises a metamaterial with an effective dielectricconstant less than 1 for at least a portion of a finite frequency range.29. A method comprising: identifying a target functionality for adielectric structure of an electromagnetic mode converting structure,wherein the target functionality comprises at least modifying a fieldpattern of a device from a first mode to a second mode; identifyingdimensions to enclose the electromagnetic mode converting structure;identifying a distribution of dielectric constants of the dielectricstructure configured to modify the field pattern to an output fieldpattern that approximates the target functionality; and generating theelectromagnetic mode converting structure where the dielectric structurehas a plurality of sub-wavelength voxels, wherein each voxel has amaximum dimension that is less than a wavelength for a specificfrequency range, and each voxel is assigned one of a plurality ofdielectric constants to approximate the distribution of the dielectricconstants.
 30. The method of claim 29, wherein the dielectric structureincludes real and imaginary parts, each of the real and imaginary partsbeing individually optimizable variables.
 31. The method of claim 30,wherein real parts are greater than or equal to approximately one andthe imaginary parts are equal to approximately zero.
 32. The method ofclaim 29, wherein the approximation of the distribution of thedielectric constant further comprises modification of at least onedielectric constant and determine a cost function for the modification.33. The method of claim 29, wherein the approximation the distributionof the dielectric constants further comprises a holographic solution.34. The method of claim 29, wherein the distribution of dielectricconstants is a three-dimensional coordinate systems.
 35. The method ofclaim 34, wherein the three-dimensional coordinate systems is one of aCartesian coordinate system or a cylindrical coordinate system.
 36. Themethod of claim 29, wherein the first mode is a TE₀₁ mode and the secondmode is a TM₁₁ mode.
 37. The method of claim 29, wherein the first modeis a TE₁₀ mode and the second mode is a TM₁₁ mode.
 38. The method ofclaim 29, wherein the first mode is a TE₁₁ mode and the second mode is aTM₁₁ mode.
 39. The method of claim 29, wherein the dimensions to enclosethe electromagnetic mode converting structure are approximate dimensionsof the device.
 40. The method of claim 29, further comprises ametamaterial with an effective dielectric constant less than 1 for atleast a portion of the finite frequency range.